8.2 Continuous Beams (Part I)

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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

8.2 Continuous Beams (Part I)
This section covers the following topics.

Analysis

Incorporation of Moment due to Reactions

Pressure Line due to Prestressing Force

Introduction
Beams are made continuous over the supports to increase structural integrity. A
continuous beam provides an alternate load path in the case of failure at a section. In
regions with high seismic risk, continuous beams and frames are preferred in buildings
and bridges. A continuous beam is a statically indeterminate structure.

The advantages of a continuous beam as compared to a simply supported beam are as
follows.
1) For the same span and section, vertical load capacity is more.
2) Mid span deflection is less.
3) The depth at a section can be less than a simply supported beam for the same
span. Else, for the same depth the span can be more than a simply supported
beam.
⇒ The continuous beam is economical in material.
4) There is redundancy in load path.
⇒ Possibility of formation of hinges in case of an extreme event.
5) Requires less number of anchorages of tendons.
6) For bridges, the number of deck joints and bearings are reduced.
⇒ Reduced maintenance

There are of course several disadvantages of a continuous beam as compared to a
simply supported beam.
1) Difficult analysis and design procedures.
2) Difficulties in construction, especially for precast members.
3) Increased frictional loss due to changes of curvature in the tendon profile.
4) Increased shortening of beam, leading to lateral force on the supporting columns.
5) Secondary stresses develop due to time dependent effects like creep and
shrinkage, settlement of support and variation of temperature.
Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

6) The concurrence of maximum moment and shear near the supports needs
proper detailing of reinforcement.
7) Reversal of moments due to seismic force requires proper analysis and design.
Intermediate spanEnd span
Intermediate spanEnd span

(a) Continuous beam in a building frame
Intermediate spanEnd span
Intermediate spanEnd span

(b) Continuous beam in a bridge
Figure 8-2.1 Continuous beams in buildings and bridges

8.2.1 Analysis

The analysis of continuous beams is based on elastic theory. This is covered in text
books of structural analysis. For prestressed beams the following aspects are
important.
1) Certain portions of a span are subjected to both positive and negative moments.
These moments are obtained from the envelop moment diagram.
2) The beam may be subjected to partial loading and point loading. The envelop
moment diagrams are developed from “pattern loading”. The pattern loading
refers to the placement of live loads in patches only at the locations with positive
or negative values of the influence line diagram for a moment at a particular
location.
3) For continuous beams, prestressing generates reactions at the supports. These
reactions cause additional moments along the length of a beam.
Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

The analysis of a continuous beam is illustrated to highlight the aspects stated earlier.
The bending moment diagrams for the following load cases are shown schematically in
the following figures.
1) Dead load (DL)
2) Live load (LL) on every span
3) Live load on a single span.
Moment diagram for DL
w
DL
Moment diagram for DL
w
DL
w
DL

w
LL
Moment diagram for LL on every span
w
LL
w
LL
Moment diagram for LL on every span

w
LL
Moment diagram for LL on one span
w
LL
Moment diagram for LL on one span

Figure 8-2.2 Moment diagrams for dead and live loads

Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

For moving point loads as in bridges, first the influence line diagram is drawn. The
influence line diagram shows the variation of the moment or shear for a particular
location in the girder, due to the variation of the position of a unit point load. The vehicle
load is placed based on the influence line diagram to get the worst effect. An influence
line diagram is obtained by the Müller-Breslau Principle. This is covered in text books
of structural analysis.

IS:456 - 2000, Clause 22.4.1, recommends the placement of live load as follows.
1) LL in all the spans.
2) LL in adjacent spans of a support for the support moment. The effect of LL in the
alternate spans beyond is neglected.
3) LL in a span and in the alternate spans for the span moment.

The envelop moment diagrams are calculated from the analysis of each load case and
their combinations. The analysis can be done by moment distribution method or by
computer analysis.

In lieu of the analyses, the moment coefficients in Table 12 of IS:456 - 2000 can be
used under conditions of uniform cross-section of the beams in the several spans,
uniform loads and similar lengths of span.

The envelop moment diagrams provide the value of a moment due to the external
loads. It is to be noted that the effect of prestressing force is not included in the envelop
moment diagrams. The following figure shows typical envelop moment diagrams for a
continuous beam.
M
max
M
min
M
max
M
min
M
max
M
min

Figure 8-2.3 Envelop moment diagrams for DL + LL

Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

In the above diagrams, M
max
and M
min
represent the highest and lowest values
(algebraic values with sign) of the moments at a section, respectively. Note that certain
portions of the beam are subjected to both positive and negative moments. The
moment from the envelop moment diagrams will be represented as the M
0
diagram.
This diagram does not depend on whether the beam is prestressed or not.

8.2.2 Incorporation of Moment Due to Reactions

As mentioned before, for continuous beams prestressing generates reactions at the
supports. The reactions at the intermediate supports cause moment at a section of the
continuous beam. This moment is linear between the supports and is in addition to the
moment due to the eccentricity of the prestressing force. The concept is explained by a
simple hypothetical two-span beam in the following figure. The beam is prestressed
with a parabolic tendon in each span, with zero eccentricity of the CGS at the supports.

The moment diagram due to the eccentricity of the prestressing force and neglecting the
intermediate support is denoted as the M
1
diagram. This diagram is obtained as M
1
=
Pe, where, P is the prestressing force (P
0
at transfer and P
e
at service) and e is the
eccentricity of the CGS with respect to CGC. Neglecting the variation of P along the
length due to frictional losses, the value of M
1
is proportional to e. Hence, the shape of
the M
1
diagram is similar to the cable profile.













Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

M
1
diagram
e
Pe
l l
Profile of the CGS
M
1
diagram
e
Pe
l l
Profile of the CGS

w
10wl / 8
3wl / 8
3wl / 8
Free body diagram of concrete
w
10wl / 8
3wl / 8
3wl / 8
w
10wl / 8
3wl / 8
3wl / 8
Free body diagram of concrete

M
2
diagram
wl
2
/2 = 4Pe
5wl
2
/8 = 5Pe
+
=
w
Pe
Simplified free body diagram
M
2
diagram
wl
2
/2 = 4Pe
5wl
2
/8 = 5Pe
+
=
w
Pe
Simplified free body diagram

Figure 8-2.4 Moment diagram due to prestressing force for a two-span beam

Next, the moment diagram due to the prestressing force and including the effect of the
intermediate support is denoted as the M
2
diagram. This is obtained by structural
analysis of the continuous beam subjected to the upward thrust. Since the profile of the
tendon is parabolic in each span, the upward thrust is uniform and is given as w
up
= w =
8Pe/l
2
. The downward thrust at the location of the central kink is not considered as it
directly goes to the intermediate support. The hold down force at the intermediate
support neglecting the downward thrust is 10w
up
l/8 = 10Pe/l. The downward forces at
the ends are from the anchorages. The moment diagram due to w
up
alone (without the
Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

support) is added to that due to the hold down force. The resultant M
2
diagram is
similar to the previous M
1
diagram, but shifted linearly from an end support to the
intermediate support.

For a general case, the resultant moment (M
2
) at a location due to the prestressing
force can be written as follows.
M
2
= M
1
+ M
1
/

(8-2.1)

In the above equation,
M
1
= moment due to the eccentricity of the prestressing force neglecting the
intermediate supports
= P
e
e.
M
1
/

= moment due to the reactions at intermediate supports.
P
e
= effective prestress
e = eccentricity of CGS with respect to CGC.
M
1
is the primary moment and M
1
/

is the secondary moment.

The moment due to the external loads (M
0
) that is obtained from the envelop moment
diagrams is added to M
2
to get the resultant moment (M
3
) at a location.
M
3
= M
2
+ M
0
M
3
= M
1
+ M
1
/

+ M
0


(8-2.2)



The variation of M
3
along the length of the beam (M
3
diagram) can be calculated as
follows.
1) The M
0

diagram is available from the envelop moment diagram.
2) Plot M
1
diagram which is similar to the profile of the CGS. The variation of P
e

along the length due to friction may be neglected.
3) Plot the shear force (V) diagram corresponding to the M
1
diagram from the
relationship V = dM
1
/dx.
4) Plot the equivalent load (w
eq
) diagram corresponding to the V diagram from the
relationship w
eq
= dV/dx. Note, over the supports w
eq
can be downwards. Also,
a singular moment needs to be included at an end when the eccentricity of the
CGS is not zero at the end.
Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

5) Calculate the values of M
2

for the continuous beam (with the intermediate
supports) subjected to w
eq
using a method of elastic analysis (for example,
moment distribution or computer analysis). Plot the M
2
diagram.
6) The M
3
diagram can be calculated by adding the values of M
2
and M
0
diagrams
along the length of the beam.

The following figures explain the steps of developing the M
2
diagram for a given profile
of the CGS and a value of P
e
.
Given profile of the CGS
e
Given profile of the CGS
e
e

Step (3) Plotting of V diagram
Step (2) Plotting of M
1
diagram
Step (3) Plotting of V diagram
Step (3) Plotting of V diagram
Step (2) Plotting of M
1
diagram
Step (2) Plotting of M
1
diagram

Step (4) Plotting ofw
eq
diagram
Step (5) Plotting ofM
2
diagram
Step (4) Plotting ofw
eq
diagram
Step (4) Plotting ofw
eq
diagram
Step (5) Plotting ofM
2
diagram
Step (5) Plotting ofM
2
diagram

Figure 8-2.5 Development of the moment diagram due to prestressing force



Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

The important characteristics of the diagrams are as follows.
1) A positive eccentricity of the CGS creates a negative moment (M
1
) and an
upward thrust.
2) The M
2
diagram has a similar shape to the M
1
diagram, which is again similar to
the profile of the CGS. This is because the moment generated due to the
reactions (M
1
/
) is linear between the supports.

8.2.3 Pressure Line due to Prestressing Force

The pressure line (thrust line or C-line) due to the prestressing force only can be
determined from the M
2
diagram. It is to be noted that the external loads are not
considered in this pressure line. This is used to select the profile of the CGS.

The calculation of pressure line from the M
2
diagram is based on the following
expression. The pressure line can be plotted for the different values of M
2
along the
length.
e
c
= M
2
/P
e
(8-2.3)
Here,
e
c
= distance of the pressure line from the CGC at a location. A positive
value of e
c
corresponds to a hogging value of M
2
and implies that the
pressure line is beneath the CGC.

The following sketch shows the pressure line for a given profile of the CGS.
e
c
e
Pressure line
Profile of the CGS
CGC
e
c
e
Pressure line
Profile of the CGS
CGC

Figure 8-2.7 Pressure line for a continuous beam

The important characteristics of the pressure line are as follows.
1) The shift of the pressure line from the profile of the CGS is a linear
transformation. It is because M
2
diagram has a similar shape to the profile of the
CGS.
Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

⇒ The pressure line will have the same intrinsic shape as the profile of the CGS.
2) Since M
2
is proportional to the prestressing force, the eccentricity of the pressure
line (e
c
) remains constant even when the prestressing force drops from the initial
value P
0
to the effective value P
e
.
⇒ The location of the pressure line for a given profile of the CGS is fixed,
irrespective of the drop in the prestressing force.

Example 8-2.1

The profile of the CGS for a post-tensioned beam is shown in the sketch. Plot the
pressure line due to a prestressing force P
e
= 1112 kN.
BA CD 7
.5 m
7.5 m6 m9 m
0.06 0.24
0.08 rad
0.12
0.176 rad
CGC
0.27
Values of eccentricity in metres.
BA CD 7
.5 m
7.5 m6 m9 m
0.06 0.24
0.08 rad
0.12
0.176 rad
CGC
0.27
Values of eccentricity in metres.

Solution

1) Plot M
1
diagram

The values of M
1
are calculated from M
1
= P
e
e.
e (m)
M
1
(kN m)
0.06
– 66.7
0.24
– 266.9
– 0.12
133.4
0.27
– 300.2

Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

133.4
– 300.2– 266.9
– 66.7
M
1
diagram (kN m)
Profile of the CGS
0.06 0.24
0.12
0.27
A D B
C
133.4
– 300.2– 266.9
– 66.7
133.4
– 300.2– 266.9
– 66.7
M
1
diagram (kN m)
Profile of the CGS
0.06 0.24
0.12
0.27
A D B
C
0.06 0.24
0.12
0.27
A D B
C

2) Plot V diagram
For AD,
1
-266.9- (-66.7)
=
9
=-22.2 kN
dM
V =
dx




For DB,
1
133.4- (-266.9)
=
6
= 66.7 kN
dM
V =
dx




For BC, to find dM
1
/dx, an approximate parabolic equation for the M
1
diagram can be
used.
( )
( )
e
e
P ex
M = - L- x
L
dM
V =
dx
P e
= - L- x
L
1
2
1
2
4
4
2





P
e
e
M
1
L
x
P
e
e
M
1
L
x





Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

At B,
1
=0
4
=-
4×(133.4+300.2)
=-
15
=-115.6 kN
x
e
dM
V =
dx
P e
L






The exact value of V at B is

V = - 107.0 kN

The difference of V between C and B is given from the change in slope of the M
1

diagram.
V|
C
- V|
B
= 0.176 × 1112
= 195.7 kN

Therefore, value of V at C is given as follows.
V|
C
= 195.7 – 107.0
= 89.0 kN
133.4
– 300.2– 266.9
– 66.7
M
1
diagram (kN m)
66.7
– 22.2
– 107.0
89.0
V diagram (kN)
133.4
– 300.2– 266.9
– 66.7
M
1
diagram (kN m)
133.4
– 300.2– 266.9
– 66.7
M
1
diagram (kN m)
66.7
– 22.2
– 107.0
89.0
V diagram (kN)
66.7
– 22.2
– 107.0
89.0
V diagram (kN)

3) Plot equivalent load (w
eq
) diagram

Include moment 66.7 kN-m at A.

Point load at D
W|
D
= 66.7- (- 22.2 )
= 88.9 kN
Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

Since B is a reaction point, the downward load at B need not be considered.
Distributed load within B and C

BC
w
89.0- (-107.0)
=
15
=13.0 kN/m


66.7
– 22.2
– 107.0
89.0
V diagram (kN)
13.0 kN/m
88.9 kN
66.7 kN m
Equivalent load diagram
66.7
– 22.2
– 107.0
89.0
V diagram (kN)
66.7
– 22.2
– 107.0
89.0
V diagram (kN)
13.0 kN/m
88.9 kN
66.7 kN m
Equivalent load diagram
13.0 kN/m
88.9 kN
66.7 kN m
Equivalent load diagram


4) Plot the M
2
diagram.
Calculate moment at supports by moment distribution
2
2
88.9×9×6
15
=128
DF
FEM
Bal
CO
Bal
Total
2
2
88.9×9 ×6
15
=-192
0.5 0.5
2
13.0×15
12
=244
–194.7
–244
–97 122
244
–38.5 –38.5
–66.7 –327.5 327.5
0
66.7
88.9
13.0
2
2
88.9×9×6
15
=128
DF
FEM
Bal
CO
Bal
Total
2
2
88.9×9 ×6
15
=-192
0.5 0.5
2
13.0×15
12
=244
–194.7
–244
–97 122
244
–38.5 –38.5
–66.7 –327.5 327.5
0
66.7
88.9
13.0

In the previous table,
Bal = Balanced
CO = Carry Over moment
DF = Distribution Factor
FEM = Fixed End Moment.

Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon










Indian Institute of Technology Madras

The moment at the spans can be determined from statics. But this is not necessary as
will be evident later.
13.0 kN/m
88.9 kN
66.7 kN m
Equivalent load diagram
327.0
– 66.7
0.0
M
2
diagram (kN m)
13.0 kN/m
88.9 kN
66.7 kN m
Equivalent load diagram
13.0 kN/m
88.9 kN
66.7 kN m
Equivalent load diagram
327.0
– 66.7
0.0
M
2
diagram (kN m)
327.0
– 66.7
0.0
M
2
diagram (kN m)

5) Calculate values of e
c
at support.

The values of e
c
are calculated from e
c
= M
2
/P
e
.

M
2
(kN m)
e
c
(m)
– 66.7
0.06
327.0
0.294
0.0
0.184

The deviations of the pressure line from the CGS at the spans can be calculated by
linear interpolation.
Pressure line
0.06
0.136
0.294
0.184
Profile of CGS
Pressure line
0.06
0.136
0.294
0.184
Profile of CGS